Saya menunjukkan kartu anggota saya kepada pegawai perpustakaan.

Yes, in a clear context you could say
• Saya menunjukkan kartu anggota kepada pegawai perpustakaan.
But dropping saya means you no longer explicitly mark possession. If it’s obviously your own card, it’s fine; otherwise it’s clearer to keep kartu anggota saya.

Kepada is the preposition for indicating the recipient of an action when that recipient is a person or group.
• ke is more for physical direction (“to” a place).
• kepada is “to” in the sense of “to someone.”
Hence, menunjukkan ... kepada pegawai perpustakaan = “show [it] to the library staff.”

• Pustakawan is the formal term for a librarian, often implying professional qualification.
• Pegawai perpustakaan is more general—any library employee or staff member, not necessarily a certified librarian.

Default Indonesian word order is SVO for transitive verbs:
Subject (Saya) – Verb (menunjukkan) – Object (kartu anggota saya) – Indirect Object (kepada pegawai perpustakaan).
You can rearrange for emphasis, e.g.
• Saya menunjukkan kepada pegawai perpustakaan kartu anggota saya,
but that places extra focus on pegawai perpustakaan.

You can turn it into passive with the prefix di- on the verb:
• Kartu anggota saya ditunjukkan kepada pegawai perpustakaan.
= “My membership card was shown to the library staff.”

Yes: memperlihatkan also means “to show” or “to let someone see.” It’s me- + per- + lihat (to see) + -kan. Both are common, but they have slightly different nuances:
• menunjukkan often implies pointing something out,
• memperlihatkan emphasizes making something visible.

While English is SVO (“I show my membership card to the librarian”), Indonesian follows the same SVO pattern, so the literal translation is fine. Just remember:
• Possessive pronouns come after the noun in Indonesian,
• Use kepada for “to [someone],”
• And the verb must carry me-/-kan to be properly transitive.